On spherically symmetric Finsler metrics with some non-Riemannian curvature properties

Abstract

In this paper, we prove that a spherically symmetric Finsler metric is a stretch metric if and only if it is R-quadratic. This yields an extension of Sadeghzadeh’s result that characterized the R-quadratic spherically symmetric Finsler metric. Then we find the necessary and sufficient condition under which a spherically symmetric Finsler metric is a weakly stretch metric. Finally, we study one of the important open problems in Finsler geometry presented by Matsumoto–Shimada about the existence of a L-reducible Finsler metric which is not C-reducible.